Compared to the behaviors of \(V_R\)  and \(V_Z\) , the vertical structure of the azimuthal velocities (\(V_\phi \) ) is more regular, with a negative \(V_\phi \) -\(|Z|\)  trend on both sides of the disk ([1]}; [2]}; [3]}; [4]}; [5]}; [6]}). The vertical gradient in the azimuthal velocities is dependent on the tracers. For instance, sampling solar neighborhood dwarf stars in the range of \(7<R<9\,\mathrm {kpc}\)  and \(|Z|<2\,\mathrm {kpc}\) , [1]} found that the gradient of the mean \(V_\phi \)  with respect to \(|Z|\)  is around \(-15\)  to \(-40\,\mathrm {km\,s^{-1}\,kpc^{-1}}\) . Using 130043 F/G-type dwarf stars within \(6.5<R<9.5\,\mathrm {kpc}\)  and \(0.1<|Z|<3\,\mathrm {kpc}\)  from the LAMOST survey and the SDSS, [5]} obtained the gradient as \(-18.5\,\mathrm {km\,s^{-1}\,kpc^{-1}}\)  for the metallicity range \(\rm {[Fe/H]}>-0.1\)  and \(-14.2\,\mathrm {km\,s^{-1}\,kpc^{-1}}\)  for \(-0.8<\rm {[Fe/H]}<-0.6\) . In most of the previous research, the patterns of \(V_\phi \)  were approximately symmetric about the Galactic plane. However, some small-amplitude asymmetric structures have been detected in recent studies. Based on proper motions of the red clump giants from the PPMXL survey [9]}, [10]} found that the rotation speed of the south disk is higher than that in the north in the range of \(8<R<13\,\mathrm {kpc}\) , though with large error bars. [11]} claimed that the K giants within \(R\geqslant 10\,\mathrm {kpc}\)  and \(Z\geqslant 0.5\,\mathrm {kpc}\)  exhibit larger rotation speed in the south disk than in the north, while at \(R\approx 8-9\,\mathrm {kpc}\)  and \(|Z|\leqslant 0.5\,\mathrm {kpc}\)  northern stars are rotating faster than the southern. In the work of [12]}, the authors found that dwarfs at \(-0.5<Z<0\,\mathrm {kpc}\)  are rotating slightly faster than those at \(0<Z<0.5\,\mathrm {kpc}\) . The faster-rotating of the southern stars was also evidenced by [13]}. They claimed that the rotation of \(Z<0\)  stars leads that of \(Z\ge 0\)  stars at \(R\gtrsim 11\)  kpc, typically by up to \(10\,\mathrm {km\,s}^{-1}\) .
